How are connection weights "perturbed"?
Perturbed here means adding a small random value to the weight. That random value comes from a uniform distribution or from a gaussian (or any distribution really). 
Imagine just nudging the weight by a little. 

It’s done to overcome the problem of local minima where models can get stuck with a good set of weights but not the best set of weights. By perturbing the weights a little, the model has a chance of finding a better set of weights. Searching for methods that deal with local minima like stochastic gradient descent will give you a better intuition.